Classical inequality measures such as the Gini index are often used to describe the sparsity of the distribution of a certain feature in a population. It is sometimes also used to compare the inequalities between some subpopulations, conditioned on certain values of the covariates. The concept of measuring inequality in subpopulation was described in the literature and it is strongly related to the decomposition of the Gini index. In this paper, the idea of conditional inequality measures is extended to the case where covariates are continuous. Curves of conditional inequality measures are introduced, especially, the curves of the conditional quantile versions of the Zenga and $D$ indices are considered. Various methods of their estimation based on quantile regression are presented. An approach using isotonic regression is used to prevent quantile crossing in quantile regression. The accuracy of the estimators considered is compared in simulation studies. Furthermore, an analysis of the growth in salary inequalities with respect to employee age is included to demonstrate the potential of conditional inequality measures.

翻译：经典的不平等度量（如基尼指数）常用于描述总体中某一特征的分布稀疏性。有时也用于比较在协变量取特定值时，某些子总体之间的不平等程度。子总体不平等度量的概念在文献中已有描述，且与基尼指数的分解密切相关。本文将该思想推广至协变量为连续变量的情形。引入了条件不平等度量曲线，特别考虑了Zenga指数和$D$指数的条件分位数版本曲线。提出了多种基于分位数回归的估计方法。为防止分位数回归中的分位数交叉，采用了基于等渗回归的估计方法。通过模拟研究比较了所考虑估计量的精度。此外，通过分析员工薪资不平等随年龄增长的变化，展示了条件不平等度量的应用潜力。